state dependent travel time models and properties

Physica A 395 (2014) 560–579

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Physica A

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M/G/c/c state dependent travel time models and propertiesJ. MacGregor Smith a,∗, F.R.B. Cruz b

a Department of Mechanical and Industrial Engineering, University of Massachusetts Amherst, MA 01003, United Statesb Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 - Belo Horizonte - MG, Brazil

h i g h l i g h t s

• M/G/c/c models provides a quantitative foundation for three-phase traffic flow theory.• M/G/c/c travel time function is S-shaped (i.e. logistics curve).• The S-shaped curve captures the phase transition from Free Flow to Congested Flow.

a r t i c l e i n f o

Article history:Received 6 August 2013Received in revised form 30 September2013Available online 5 November 2013

Keywords:TransportationState dependent queueTravel timeApproximations

a b s t r a c t

One of the most important problems in today’s modeling of transportation networks isan accurate estimate of travel time on arterial links, highway, and freeways. There are anumber of deterministic formulas that have been developed over the years to achieve asimple and directway to estimate travel times for this complex task. Realistically, however,travel time is a random variable. These deterministic formula are briefly reviewed andalso a new way to compute travel time over arterial links, highway, and freeways, ispresented based on an analytical state dependent queueing model. One of the features ofthe queueing model is that it is analyzed within the context of the theoretical three-phasetraffic flowmodel. We show that the model provides a quantitative foundation alternativeto qualitative three-phase traffic flow theory. An important property shownwith themodelis that the travel time function is not convex, but a sigmoid S-shaped (i.e. logistic curve).Extensive analytical and simulation experiments are shown to verify the S-shaped natureof the travel time function and the use of the model’s method of estimation of travel timeover vehicular traffic links as compared with traditional approaches. Finally, it is shownthat the point-of-inflection of the S-shaped curve represents the threshold point wherethe traffic flow volume switches from Free Flow to Congested Flow.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

‘‘Therefore, queueing theories cannot be relied upon for a correct description of congestion on un-signalized freeways, and theterm ‘queue’ will henceforth not be used to describe traffic on freeways’’, Boris S. Kerner, The Physics of Traffic, p. 85, Springer-Verlag, Berlin, 2004.

Most design and analysis studies of transportation networks need to estimate in some way the travel time over thetraffic network segments. This is a fundamental but difficult problem because of congestion, natural variability of ratesof travel, time of day, road and weather conditions, driver characteristics, and vehicle types. The traffic flow process isessentially a stochastic (random) process. Since this is a random process, one needs to estimate the probability distributionof the number of vehicles along a roadway in order to compute the desired performance measures of the roadway such asthroughput, queueing delays, average number of vehicles along the segment and the utilization of the roadway segment.

∗ Corresponding author. Tel.: +1 14135454542.E-mail addresses: [email protected], [email protected] (J. MacGregor Smith), [email protected] (F.R.B. Cruz).

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It is important to have an accurate formula since the use of this formula is critical to transportation planning and trafficassignment activities.

1.1. Motivation and purpose

In this paper, we illustrate the special properties of a state dependent queueing approach to traffic modeling. Part of thereason for this is to show that through the queueing approach, it helps explain the often frustrating experience that driversface when there is severe congestion on roadways. On another account, with the queueing approach, it begins to explainquantitatively some of the fundamental assertions of Kerner’s three-phase traffic theory [1].

Another feature of this paper is to develop an approach to understanding what type of travel time calculations will resultfrom the queueing approach. This is motivated by the following situations and examples.

For example, in estimating the amount of fire insurance for a community’s fire protection, insurance companies oftenutilize a simple formula to estimate the response time for an engine company to respond to a fire. While the fundamentalresponse time is a stochastic process, the insurance companies will often use a deterministic formula to estimate thisresponse time. In fact, one formula often used is based upon linear-regression research by Hausner et al. [2] and is thefollowing:

T = 0.66 + 1.77D,

where:

T := time in minutes to the nearest 1/10 of a minute;0.66 := a vehicle acceleration constant for the first 1/2 mile traveled;1.77 := a vehicle speed constant validated for response distances ranging from 1/2 miles to 8 miles;D := Euclidean distance based upon two-dimensional Cartesian coordinates from the engine company location to thefire location.

This formula was derived in a RAND Corporation study in New York city and is still used as a predictive tool. The formulais somewhat dated and is derived from empirical data in Yonkers, New York, but appears to be a very practical one [3].

Another situation is in the traffic planning area, which is considered a crucial component since stable transportationsystems are one of the main factors that contribute to a higher quality of life. Indeed, Buriol et al. [4] propose an algorithmto solve a user equilibriummodel (which describes the behavior of users on a given traffic network), while maintaining thesystem optimummodel solution (which describes a traffic network operating at its best operation). This algorithm is basedon a convex approximation to represent the cost of traveling along each arc, as a function of the flow on the arc (more ontraffic assignment models on the classical book of Sheffi [5]). This is a relevant area of research since any improvement issignificant because of the many millions of dollars that are spent every day on traffic issues [6,7].

There have been many formulas developed over the years for estimating roadway traffic travel times. Why is there anyneed for another formula? First of all, it is important that the theoretical formula reflects as closely as possible what happensin reality. If the understanding of the travel time phenomenon is not empirically and theoretically sound, then all themodelsbased on the formula are deficient.

Secondly, if one can easily compute a more accurate estimate of travel time in congested environments, then this will bean important boost to its application in traffic modeling and traffic network design. As might be expected, one does not getsomething from nothing. In order to achieve a more accurate estimate, the tradeoff is that more computation work has tobe done, yet the real benefit here will be a better approximation to the quantitative measure for congested highway traffic.With the advent and proliferation of powerful computer processors (even in cell phones and their web inter-connectivity),the computation times should not become extraordinary.

1.2. Outline of paper

Section 2 presents a survey of relevant literature about travel time models and discusses two-phase and three-phasetraffic theory of Kerner [1] in contrast to the theory which derives from the state dependent approach. Section 3 presentsthe state dependent model and its derivation and computation of travel times along highway segments. Section 4 presentsa comparison of the computation of travel times in particular the Bureau of Public Roads [8] formula and a formula byAkçelik [9] along with the state dependent model. Section 5 examines some incidents and bottlenecks and compares thestate dependent approach with the other travel time models. Section 6 summarizes and concludes the paper.

2. Problem and literature review

In traffic assignment and evacuation planning models, a travel time delay function is necessary in order to express therelationship between the traffic flow volume and the expected delay on the traffic link. Numerous formulas have beenconstructed over the past 40 years for this purpose, among some of them are the contributions by the following authors:Bureau of Public Roads [8]; Davidson [10]; Rose et al. [11]; Spiess [12]; Akçelik [9]; and Dowling et al. [13]. From a physics-oriented approach, there is the thorough review by Chowdhury et al. [14] in which the authors explain the detailed basic

562 J. MacGregor Smith, F.R.B. Cruz / Physica A 395 (2014) 560–579

Fig. 1. Correlation between flow rate Q and vehicle density ρ and the two-phase traffic flow model (free flow and congested flow) for various empiricalstudies provided by Kerner [1].

principles behind the main theoretical approaches from a statistical physics perspective. In the review by Helbing [15],empirical data are considered and the main approaches to modeling pedestrian and vehicle traffic are reviewed includingthe microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Finally, Nagatani [16]presents various traffic models, explains some of the physical phenomena associated with traffic, discusses analytical andnumerical techniques for analyzing these models, and presents results focused on the microscopic car-following models.There are even other papers, but the above sample is considered representative.

2.1. Criteria of formula

In one of the previous papers, Spiess [12] has recommended a series of seven guidelines for awell-behaved delay function.If one posits a function f (x) where x represent the ratio of volume to capacity v/k then Spiess recommends the followingproperties the function should maintain [17].

1. f (x) should be strictly monotone increasing.2. f (0) = 1 and f (1) = 2. In other words, the function should yield the free flow travel time at zero volume and twice the

free flow travel time at capacity.3. f ′(x) should exist and be strictly monotone increasing, i.e. the derivative is representative of a convex function. A convex

function is desirable for optimization purposes.4. f ′(1) = α, the exponent in the BPR function, i.e. the function has only a few well-defined parameters.5. f ′(x) < Mα, whereM is a finite positive constant, i.e. the function should be finite for all volumes.6. f ′(0) > 0, the derivative is positive at zero volume.7. The evaluation of f (x) should not take more computation time than the evaluation of the corresponding BPR function.

These guidelines are interesting because they can be used to compare alternative formulas. There is some argument thatcriterion #2 should be relaxed [17], since there is no intuitive rationale for such a criterion besides to ensure compatibilitywith thewell known BPR type functions [12], and that the third and seventh standards are inhibiting and should be dropped.The seventh criterion is important since if the calculations are excessive, then it will create convergence problems in thetraffic assignment calculations.

Before we present the state dependent queueing model used to compute travel time, let us review the basic way ofincorporating congestion in traffic flow models through a two-phase traffic flow model, then a three-phase traffic flowmodel.

2.2. Two-phase traffic models

In two-phase traffic flow models, the argument is that there are only two phases of vehicular traffic flow: (i) Free Flowand (ii) Congested Flow. There are empirical data which show a strong correlation between the flow rate Q and vehicledensity ρ, so that there is an upper boundary at the maximum point of free flow that corresponds to also a critical densityvalue (see [1], pages 22–26]. This two-phase traffic flow is illustrated in Fig. 1 which is based upon the diagram in Kerner’svolume [1]. The empirical data points are provided by Kerner [1] from studies in roadways. Some of the formula based upontwo-phase traffic flow theory are reviewed in Section 4.


Fig. 2. Relationship between the flow rate Q , the vehicle density ρ, and Kerner’s three-phase flow theory (free-flow, F , and synchronized-flow, S).

2.3. Three-phase traffic models

As we shall argue, three-phase traffic flow models originate with Kerner’s research [1] and the proposition that three-phases occur in traffic modeling, so that in Congested Flow there are two additional phases, Synchronized Flow and Wide-Moving Jam Flow as follows.

(i) Free Flow:(1) Free Flow Traffic;

(ii) Congested Flow:(2) Synchronized Flow;(3) Wide-Moving Jam Flow.

One of the features of Kerner’s work [1] is that he only seems to be interested in the empirical foundation for thethree-phase traffic flow models. This notion of phase transitions as we shall argue is similar to what happens for thestate dependent model. Notice that we do not mean to imply that the three-phase models are widely accepted and thatconsistency with them lends credence to our state dependent model. Notable studies along these lines include Schónhofand Helbing [18] and Treiber et al. [19], which calls the theory into question on theoretical and empirical grounds. Theseare studies regarding travel time estimation in the context of considerably recent and on-going research on travel timemonitoring.

One of Kerner’s main hypotheses is that there is an infinite number of highway capacities of free flow at a bottleneck.These infinite capacities are bounded between a minimum andmaximum capacity (see Kerner [20], chapter 4). We have noproblem with this argument, however, we want to show that the phase transitions can occur without a bottleneck.

Fig. 2 illustrates the relationship between the flow rate of vehicular traffic and the density of vehicles in the three-phasetraffic flow theory. This diagram with the appropriate filled color areas is modeled after Kerner’s book (see Kerner [20],p. 76). F → S illustrates the free flow (F ) until there is some bottleneck at which the synchronized flow (S) occurs. Noticein red the F → S transition. Within the S phase, self compression of the vehicles comes into play and the density increasesand this self compression is due to what Kerner calls the ‘‘pinch effect’’. There are minimum and maximum capacities (i.e.throughputs) which then occur. In fact, there are infinite capacities within a range of critical capacities.

The model to be presented in Section 3 can generate the following phenomenon as shown in Fig. 3, which illustrates thesudden disruptive effects and decreasing flow volume (i.e. throughput) which can occur while driving along a roadway atthe lone occupant speed (maximum speed), V1 = 50 mph respectively for a freeway stretch of one mile and one lane widewhere the jam density is 220 vehicles/mile. Thus, as illustrated in Fig. 3, there is a halt to the monotonic increase in flowvolume (throughput), then a sudden decrease, and finally a leveling off. There is a myriad of flow rates (see the bottleneckeffect region in Fig. 3) between the crucial input arrival rate and the saturated flow volume. From a queueing perspective,this is a remarkable phenomenon. Most plots of throughput in finite buffer queueing studies are smoothmonotonic concavefunctions that reach a plateau and do not descend, whereas this throughput curve is quasi-concave at best. These phantomeffects of sudden slow-downs are also related to the empirical approach and understanding of the three-phase flow of traffictheory developed by Kerner [1].

In contrast to Fig. 2, Fig. 3 illustrates that at the ‘‘pinch point’’, there is also a sudden drop in capacity and eventuallyit levels off to a constant capacity (throughput) because the traffic has reached the jam density of the roadway segment.What Fig. 3 clearly shows is that based upon a single road segment, not even a bottleneck, the state dependent modelindicates the threshold effects of moving from free-flow (F ) to a wide moving jam (J) with an intermediary threshold stage(i.e. F → S → J). State dependent traffic flow models offer a ‘‘quantitative’’ explanation to the qualitative observations of


Fig. 3. Flow volume (throughput) for a freeway stretch of one mile long and one lane wide for a maximum speed of V1 = 50 mph and a jam density of220 vehicles/mile.

Fig. 4. Finite queueing situation created by a road segment with finite length and width together with the vehicle sizes.

Kerner’s theory since the queueing theory indicates a crucial relationship between the blocking effects of the finite queue,the arrival flow rates, and the ultimate capacity (i.e. throughput) of the finite queue. A bottleneck can induce a wide movingjam, but the wide moving jam can occur without the bottleneck! The roadway itself is a bottleneck.

Let us explain in some detail in the following section the state-dependent M/G/c/c model and its properties and whywe feel that a traffic flow theory based uponM/G/c/c queueing models is worthy of examination.

3. M/G/c/c state dependent model

We assume that travel time T is a random variable and that it is a function of the volume of traffic along the road segment.This road segment has finite length and width and these together with the vehicle sizes create a finite queueing situation,as seen in Fig. 4. Thus, it is important here in this model to account for the length of the roadway segment not only its width(# of lanes) as this will directly affect the physical capacity of the road segment. Fig. 4 illustrates four traffic lanes (roadsegments) which underly the basic relationship of the M/G/c/c approach in which M stands for a queueing system withstationary Poisson arrivals at some rate λ, G for a general service time distribution (which will be considered state dependentin a sense that will be made clear soon), c servers in parallel, and room for at most c customers including those in service(i.e. no queue).

3.1. Notation

At this stage of the paper, it is useful to have a list of important notation used in the discussion. Since the audienceis comprised of physicists, traffic researchers, and operations researchers, we try to provide the notation that is commonparlance.

M := Markovian arrivals (i.e. Poisson arrivals);G := State dependent general service time probability distribution;c := Number of servers and physical capacity of anM/G/c/c queue;L := Length of the road segment (miles or km);


W := Width of the road segment (# lanes);ρ := jam density parameter, in vehicle/mile-lane (normally ranging from 185–265 vehicle/mile-lane);Λ := External Poisson arrival rate to the network;λj := Arrival rate to internal network node j;pc := Blocking probability in an M/G/c/c queue (i.e. probability that arrivals are rejected because the queueing systemis full);Q := Capacity or vehicular flow volume (i.e. throughput of a highway segment);θ := Mean throughput rate which is equivalent to Q above;T := Travel time along a road segment or a path network;WIP := Average work-in-process or number of vehicles in the queueing system.

3.2. Basic relationships

In the basic road segment model, the maximum number of vehicles allowed on a road segment is captured by the (jamdensity) c is the following expression:

c = ρ × L × W . (1)

That all highway segments have finite capacity (which we will define as c) based upon the jam density, ρ, the geometryof the roadway (i.e. its length, L, and number of lanes,W ), and natural geography and topographywhere the roadway residesis perhaps an obvious observation. However, the implications of the finiteness are very critical for all that is to follow. Sincethere is a bounded length and width and depending upon the jam density, the highway can only handle a finite number ofvehicles. One could argue that the bounding of the length is arbitrary, yet there are natural features of a highway such aschanges in slope, plateaus and valleys, and the presence or absence of on- and off-ramps which naturally suggests a finitebound to L.

The state dependent model is a stochastic model which requires one to compute the probability distribution of thenumber of vehicles along a roadway as a function of the density of vehicular traffic traveling down the road segment. Thespeed-density curves which normally describe the vehicle speed along a segmentwill be used by the state dependentmodelto calculate the probability of the number of vehicles along the road segment. Once this probability distribution is found,the mean time to traverse the segment can be computed.

It is important to point out that the state dependentmodel is both a combinedmacro-level as well as amicro-level trafficmodel. It incorporates themacro behavior of individual cars through the state dependent curves, yet models each individualcar traveling the highway segment. Each individual vehicle along the highway segment has its speed i.e. service rate adjusted‘‘dynamically’’ to the number of vehicles traveling along the segment. As one vehicle enters or leaves the road segment, allthe other vehicles still on the segment have their speeds adjusted. Whichever speed density curve is appropriate to thehighway segment can be utilized in the state dependent model. Another important notion for the M/G/c/c model is thatthere is no queue or queue discipline. The vehicle is either on the highway segment or it is not.

Fig. 5 on the left presents an exponential approximation of empirical M/G/c/c state-dependent traffic flow modelfrom (a) Jain and MacGregor Smith [21], and empirical distributions for vehicular traffic flows from (b) Edie [22], (c)Underwood [23], (d) Greenshields [24], (e) Drake et al. [25], and (f) Transportation Research Board [26]. Fig. 5 on the rightillustrates experimental curves, (a) through (f), that relate the walking speed of pedestrians to crowd density, based onvarious empirical studies that illustrate that at a mean density of 3 pedestrians/m2 walking is reduced to a shuffle, and at5 pedestrians/m2, forward movement essentially comes to a halt, as reported by Tregenza [27]. The striking similarity tothe speed-density curves indicates that the results developed in this paper for vehicle modeling applies also to pedestriantraffic flow modeling [28]. It is felt that many other particulate flows have also this speed decaying behavior as a functionof density.

Classicalmacroscopic traffic studies byGreenshields andGreenberg have documented these linear and exponential decayfunctions for vehicular traffic (see May [29]), and the state-dependent models can capture this macroscopic behavior. Animportant observation here is that any empirical curve can be utilized in the segments of the traffic studies through thequeueing representation. One is not restricted to one single fixed speed-density curve. This yields a great deal of flexibilityfor the approach, which is not readily affordable in other approaches, except perhaps simulation [30,31]. The problem withsimulation is that it is very expensive from a computational running time and storage viewpoint to dynamically update theservice rate of each vehicle in the traffic segment as a function of density of traffic in the segment. This is not a problem forthe analytical models.

Over many years, a generalized model of theM/G/c/c Erlang loss queueing model for this service rate decay which canmodel any service rate distribution (linear, exponential, . . . ) has evolved [28,32,33]. It is a special case of an Erlang lossmodel.

3.3. Derivation of probabilities

The basic probability distribution for a linear congestion model and an exponential congestion model will be shown. Theprobability distribution is critical to the performance measures and especially the travel time function. Some of the detailsof this development are deferred to previous articles [21,28].


(a) Exponential approximation of empiricalM/G/c/c state-dependent traffic flow model from(a) Jain and MacGregor Smith [21], and empiricaldistributions for vehicular traffic flows from (b)Edie [22], (c) Underwood [23], (d)Greenshields [24], (e) Drake et al. [25], and (f)Transportation Research Board [26].

(b) Experimental curves, (a) through (f), thatrelate the walking speed of pedestrians tocrowd density, based on various empiricalstudies reported by Tregenza [27].

Fig. 5. Approximation of empirical vehicular/pedestrian speed-density curves.

3.4. Linear congestion model

The linear congestion model is based on the idea that the service rate of the servers in theM/G/c/c queueing model is alinear function of the number of occupants in the system. This is basically the approach of Greenshields (see May [29]). It isworthwhile mentioning though that linear/exponential models which do not contain the loss feature are also possible, forinstance G/G/1 and M/G/1 models [34,35].

A linear congestion model can be developed where the vehicle speed of a single vehicle is Vn and as it approaches thejam density c of the road segment Vn → 0. For this reason, since a vehicle population of n = c + 1 is impossible, Vn = 0 isset for all n ≥ c + 1.

Thus, if V1 is the average travel speed of a lone vehicle and Vc+1 = 0, then:

Vn =V1

c(c + 1 − n) . (2)

Eq. (2) gives the vehicle-speed of n vehicles in a single lane, which is also the number of busy servers. Note that the servicerate, rn, of each of n vehicles in the lane, is the average of the inverse of the time it takes these individuals to traverse thelength of the lane; therefore,

rn = Vn/L. (3)

Using Eq. (2), this gives us,

rn =V1

cL(c + 1 − n). (4)

The service rate of the queueing system (overall) is equivalent to the number of servers in operation (i.e., occupied)multiplied by the rate of each server. Since all n servers, in a state-dependent M/G/c/c queueing model, operate at thesame rate, rn, we have that µn, the overall service rate of the system when there are n vehicles in the lane, may be definedas:

µn = nrn =V1

cL(c + 1 − n)n. (5)


Expressions for the state probabilities are derived by substituting the expression for µn, Eq. (5), into the Chapman–Kolmogorov equations for solving the probabilities of a single queue

pn =λn

ni=1

µi

p0, (6)

for n = 1, . . . , c , and

1p0

= 1 +

cn=1

λnn

i=1µi

, (7)

to obtain Eqs. (8) and (9):

pn =λn

V1cL

n ni=1

(c − i + 1)ip0, (8)

1p0

= 1 +

cn=1

ℓn

ni=1

(C − i + 1)i

, (9)

where V1 is the free-flow speed, and ℓ = λcL/V1. Note that L is expressed in meter (or miles) and λ is expressed in hour−1.Thus, Eqs. (15) and (16) gives us the desired probability distribution for the linear congestion model.

3.5. Exponential congestion model

In developing the exponential congestion model, assume that, rn, the service rate of each of the n occupied servers, isrelated to the number of vehicles by an exponential function. The form of the exponential function is based on the equationfor the vehicle-speed, as depicted by the following relation.

Vn = V1 exp−

n − 1

β

γ . (10)

Parameters β and γ are found by fitting points to the curve in Fig. 5. Parameters β and γ will be referred to as the scaleand shape parameters respectively. Fitting the points (1, V1), (a, Va), and (b, Vb) gives one the algebraic relationships shownbelow:

γ = lnln (Va/V1)

ln(Vb/V1)

ln

a − 1b − 1

, (11)

β =a − 1

ln

V1Va

1/γ =b − 1

ln

V1Vb

1/γ . (12)

By carefully approximating the positions of three representative points among the curves in Fig. 5, the following samplecoordinates are utilized:

Vn = 62.5 mph ⇒ at densityδ = 1/LW vehicles/mile-lane ⇒ n = δLW = 1,

Vn = 48.0 mph ⇒ at densityδ = 20 vehicles/mile-lane ⇒ n = a = 20LW ,

Vn = 20.0 mph ⇒ at densityδ = 140 vehicles/mile-lane ⇒ n = b = 140LW .

See Jain and MacGregor Smith [21] for further details. One should select different points to fit the curve when the free-flow speed V1 changes.

Combining Eqs. (3) and (10) gives,

rn =V1

Lexp

−

n − 1

β

γ , (13)


where V1 is the free-flow speed. Therefore, one can express the overall service rate of theM/G/c/c queueing model as

µn = nrn = nV1

Lexp

−

n − 1

β

γ . (14)

Equations for the state probabilities are obtained by substituting the expression for µn, Eq. (14), into Eqs. (6) and (7) toobtain Eqs. (15) and (16):

pn =λn

ni=1

i

V1L

exp

−

i−1β

γ p0, (15)

where

1p0

= 1 +

cn=1

λn

ni=1

i

V1L

exp

−

i−1β

γ . (16)

Note that V1 can be expressed in miles per hour (mph) or kilometers per hour (km/h), L is expressed in miles or meters,and λ is expressed in hour−1.

3.6. Final remarks

For both the linear and exponential congestion models the expected delay (i.e., the expected travel time), E(T ), may befound by computing the average number in the queue, E(N), then dividing it by the arrival rate, λ. Indeed, from Little’s Law,one knows that under steady state conditions the average number of items in a queueing system equals the average rate atwhich items arrive multiplied by the average time that a item spends in the system, E(N) = λE(T ) (about Little’s Law, thereader may want to check Little and Graves [36], for a thorough discussion).

Since there exists a finite capacity queue, the effective arrival rate (i.e. excluding the items that are blocked) must beconsidered instead, which is defined in terms of the external arrival rate λ and the blocking probability pc , λ̃ = λ(1 − pc),which gives that E(N) = λ̃E(T ). Finally, since t ≡ E(T ) then one achieves:

t =E(N)

λ̃= λ̃−1

cn=1

npn. (17)

It would be most fortunate if one could achieve a closed form expression of the travel time delay formula, but this doesnot appear to be possible because t is a complex function of c in the probability calculations. One could fix c and derive sucha function, but it would be tedious to deal with all these functions since there could be thousands of such functions.

4. Expected N and T calculations

In this section of the paper, the travel time calculations of the three different models over a freeway segment areillustrated for varying lengths L = 1, 2, 5, 10 miles and variations in the traffic volumes. Let us introduce two alternativeways of computing travel time which we will contrast with theM/G/c/c approach.

4.1. Alternative ways of computing travel time

4.1.1. BPR formulaThe BPR formula [8] is probably the most well-known formula used in estimating travel delay. It was developed in 1964

using data from the Highway Capacity Manual [26]. The formula is:

t = tf [1 + α(x)β ],

where:

t := predicted travel time over the road segment;tf := travel time at the free-flow speed;v := traffic volume (synonymous with traffic flow rate);k := practical capacity (usually defined as 80% of actual capacity, in vehicles/h);x := ratio of volume to capacity, i.e. x = v/k;α := ratio of free flow speed to the speed at capacity (typically α = 0.15);β := indicates how abruptly the curve drops from the free-flow speed (typically β = 4.0).


Table 1Suggested values for JA [9].

Facility type tf Q JA(km/h) (vehicles/h)

Freeway 120 2000 0.1Expressway 100 1800 0.2Arterial 80 1200 0.4Collector 60 900 0.8Local Street 40 600 1.6

4.1.2. Criticism of the BPR formulaFirst of all, we must recognize the importance of the BPR formula [8], since it is probably the most widely used one. This

classical formula has been considered by a legion of researchers (e.g. see Prashker and Bekhor [37], Bell and Cassir [38],Braess et al. [39], Buriol et al. [4], Zheng and Liu [40], to cite a few) and it is only recently that comparisons questioningthe supremacy of the BRP formula have started to appear in the literature (see García-Ródenas et al. [41], Ghatee andHashemi [42], Cruz et al. [43]).

Since the BPR formula is an empirically based model, there have been a number of criticisms of the values of theparameters set in the model. Dowling et al. [13] recommend an updated version of the parameters. Thus, α = 0.05, forsignalized facilities, and 0.20, for all other facilities, and β = 10.0.

The resulting speed-flow curve is flatter than the original BPR curve for v/k ratios < 0.70 and the new curve drops morequickly in the vicinity of capacity v/k = 1.00. We will defer to Dowling et al. [13] for their experiments and rationale forthese adjustments.

4.1.3. Akçelick’s formulaAkçelik [9] developed his travel delay formula based on the steady-state delay equation for a single channel queueing

system. A time-dependent form of the delay equation was then arrived at using a coordinate transformation method:

t = tf +

0.25T

(x − 1) +

(x − 1)2 + 8

JAQT

x

,

where:

t := average travel time per unit distance (h/km);tf := travel time at the free-flow speed (h/km)x := ratio of volume to capacity (i.e. v/k, also called the degree of saturation);T := the flow period (hours) (typically one hour);Q := capacity or flow rate (vehicles/h);JA := a delay parameter.

The delay parameter JA ensures that the delay equation will predict the desired speed of traffic when demand is equal tocapacity. In fact, Akçelik [9] has the following equation to calculate this quantity:

JA =2QT

(tc − tf )2,

where tc is the rate of travel at capacity. The suggested values for the delay parameter are presented in Table 1.

4.1.4. RemarksNotice that neither the BPR [8] or Akçelik’s [9] model takes into account the finite length andwidth of the road segments.

We feel that this is a distinct disadvantage of these prior models. While Akçelik’s model is based on a single-channelM/G/1queue, the model that follows is actually a queueing model for the traffic road segment, not an analogous model.

In the experiments that follow, a one-lane freeway road segment was assumed with V1 = 62.5 mph ≡ 100 km/h. Thecapacity k = 2400 vehicles/h was set from the Highway Capacity Manual [26]. The BPR formula used had the modifiedparameter values α = 0.20, β = 10, and for Akçelik’s model the delay parameter and time value were set JA = 0.10 andT = 1.0 respectively. For the M/G/c/c experimental results, a jam density of 200 vehicles/mile-lane was assumed and allother parameters for the linear and exponential models were used as defined previously.

Additionally, a discrete-event digital simulation model was run to confirm the accuracy of the solutions generated. Thesimulation model was developed by Cruz et al. [30] and implemented as C++ program. The simulation model considers aroad segment as anM/G/c/c state dependent queue, with Poisson arrival rate λ, general service rate G and limited capacityc. The simulations took place on a PC, CPU Pentium II 400 MHz, 256 MB RAM, under Windows NT 4.0 operating system. Thesimulation timeswere set to 20 h,with a burn-in period of 10 h. In order to compute 95% confidence intervals, 30 replicationswere performed. Longer and shorter simulation time settings were tried but the results (not shown) were not significantlydifferent.


4.2. Linear state dependent model

As seen in Fig. 6, probably the most significant result is that the travel time function of the M/G/c/c model is S-shaped,it is not convex (see Guideline #3). This property is not necessarily bad since it means that the function is quasi-concave,but it is not convex. It is important to see from the curve that the travel time t resulting from theM/G/c/c state-dependentmodel is not monotone increasing, but actually dt

dλ → 0, i.e. the derivative of t with respect to the arrival rate λ goes to zerofor some threshold value of the arrival rate (or traffic demand) λ. This threshold value of traffic demand λ is thus a limitingvalue for the volume to capacity ratio, x = v/c . It does appear from all the curves that in the linear state dependent model,there is an upper limit near 2100 vehicles/h approximately when the vehicles will start to slow down which is consistentwith the estimated capacity c = 2400 vehicles/h. Therefore, the derivative of the travel time formula of theM/G/c/c modelgoes to zero around this point.

This leveling off of the travel time function makes sense because as the traffic volume approaches the jam density, it willslow down andmonotonically increase in value, but not stop altogether, since unless impeded by an incident, the traffic willkeep moving. Notice that average speed is meant here since even without an incident traffic has the stop-and-go behaviorin very heavy conditions.

Table 3 (in the Appendix) shows analytical and simulated performance measures for the linear model, namely theblocking probability pc , the throughput θ , the expected number of cars on the road link E(N), the expected service timeE(T ), and the cpu time to run the simulation. The confidence intervals are certainly too narrow to be noticed in the figuresbut are instructive because they are showing the low variability of the estimates and the close agreement between theM/G/c/c model and the simulation.

Fig. 7 illustrates a comparison of the linear analytical and simulation values for the 1, 2, 5, and 10 mile experimentsrespectively starting from the top left hand figure.

4.3. Exponential state dependent model

As in the linear state dependent model, the travel time curve for the exponential state dependent curve is also S-shaped.This is illustrated in Fig. 8. For the exponential model, which is perhaps more realistic for traffic segments, it does not riseso steeply and abruptly as is the case for the linear model. The linear model more abruptly reduces the speed as the usagegets closer to the capacity C , unlike the exponential model, which is smoother.

One important aspect of the exponential M/G/c/c model is that it always achieves Guideline #2 of Spiess’s guidelinesin that it predicts that the travel time is at least twice the free-flow time at capacity. Also, the M/G/c/c model is morepessimistic than the other two models, but this makes sense since the travel speed on the link is dynamically adjusted bythe speed-density curve as the traffic density increases.

As one can see, when the length of the freeway section is short, Akçelik’s model [9] is similar to the state dependentmodel but when the length of the freeway is up to 10 miles, Akçelik’s model does not capture the congestion delay. TheBPR model [8] seems to agree pretty well with the state dependent model in that the BPR curves follow closely the state-dependent model, except that they are more optimistic in the lower volumes and go off to infinity at higher volumes.

Table 4 in the Appendix shows all performance measures and their respective confidence intervals, the blockingprobability pc , the throughput θ , the expected number of cars on the road link E(N), the expected service time E(T ), and thecpu time to run the simulation.

Fig. 9 illustrates a comparison of the exponential analytical and simulation values for the 1, 2, 5 and 10mile experimentsstarting from the top left hand figure. As a result of this set of experiments, we have the first definition for calculating thetravel time in anM/G/c/c network model, which declares that:

Definition 1. For a given highway segment with parameters L, W , and other parametric factors (V1, Va, Vb, . . .), the traveltime function of theM/G/c/c model is S-shaped

T (λ) :=Tmax

1 + e−κ(λ−λ∗)(18)

where:

Tmax := maximum travel time;λ := traffic flow volume;λ∗

:= flow volume threshold at point-of-inflection;κ := parameter integrating theM/G/c/c model and the highway characteristics;

Notice that Eq. (18) is just a logistic function (or logistic curve), which is a common sigmoid curve [44]. The particularform of the S-shaped function is not unique. However, in queueing models, this form of the function is pretty useful. Thisdefinition follows from the nature of the M/G/c/c model and the experimental results demonstrated in this paper. Theexpression argues that the travel time function T (λ) is bounded and not infinite. No matter which state dependent curveis utilized, there would be maximum travel time threshold Tmax for the given highway or roadway parameters. The above


Fig. 6. Travel time curves for the linear M/G/c/c flowmodel, the BPR formula, and for the Akçelik’s model, for single-lane freeways of 1, 2, 5, and 10mileslong, clockwise from the top left.

expression is suitable to deriving this expression from sample data. However, with the software for theM/G/c/c model, theS-shaped curve can be directly generated.

Conjecture 1. The travel time function between any two points on an un-signalized highway network for the M/G/c/c model isan S-shaped function.

While the proof of this is dependent upon isolating second derivative information of theM/G/c/c function, this appearsto be a very difficult problem since theM/G/c/c travel time function is not in closed form. While the proof of this propertyofM/G/c/c functions is to be determined, it is also expected that the general travel time function on a highway network isS-shaped, it does not go off to infinity as is expected in the BPR or related deterministic formulas. Some empirical evidencefollows in the next section of the paper.

There have been many conjectures regarding the maximum flow rate Tmax for a highway. See the Highway CapacityManual [26], exhibit 8–19 for a select set of maximum flow rate capacities for various metropolitan areas. What is providedin this paper is that now this maximum travel time Tmax can be found for a particular arterial or highway segment with theunderstanding of the S-shaped travel time function and theM/G/c/c state dependent model.

Finally, and most importantly, once one has the S-shaped curve, one can calculate the point of inflection for the curveand this will indicate the threshold traffic flow volume inwhich the transition stage from Free Flow to Congested Flow occurs.The point-of-inflectionmarks the switch from the convex to the concave part of the S-shaped function. This definition againfollows from the basic properties of the S-shaped curve.

Definition 2. The point of inflection of the S-shaped travel-time curve in the M/G/c/c model is the value of the trafficvolume at which the Free Flow traffic volume switches to Congested Flow.

While it may be computationally difficult to find this point-of-inflection in the travel time function, the followingconjecture indicates a key relationship between the point-of-inflection of the travel time curve and the throughput or flowrate function Q .


Fig. 7. Travel time curves for the linear M/G/c/c flowmodel and simulations for single-lane freeways of 1, 2, 5, and 10 miles long, clockwise from the topleft.

Fig. 8. Travel time curves for the exponential M/G/c/c flow model, the BPR formula, and for the Akçelik’s model, for single-lane freeways of 1, 2, 5, and10 miles long, clockwise from the top left.


Fig. 9. Travel time curves for the exponential M/G/c/c flow model and simulations for single-lane freeways of 1, 2, 5, and 10 miles long, clockwise fromthe top left.

Fig. 10. The maximum flow rate of the throughput function (left) and the point-of-inflection (right) of the travel time function for anM/G/c/c model fora freeway stretch of one mile long and one lane wide for a maximum speed of V1 = 55 mph.

Conjecture 2. The point-of-inflection of the travel time function T (λ) corresponds to the maximum flow rate θmax of thethroughput function.

Although no formal proof appears possible at the present time, the relationship holds experimentally as is shown inFig. 10. Fig. 10 illustrates the correspondence between the throughput function and the travel time function for a freewayof 1 mile long and 1 car width wide. Roughly speaking at the flow volume of λ ≈ 2700 vehicles/h, θmax equals the point-of-inflection of T (λ). We have modeled this relationship between the throughput function and the travel time function for aseries of V1 values.

Thus, whether one ascribes to the two-phase or three-phase traffic flow theory, the point-of-inflection of the travel timefunction indicates the transition threshold for moving from Free Flow to Congested flow. From a practical point-of-view, ifone can find themaximum throughput rate, then this corresponds to the point-of-inflection value in the travel time function.On the other hand, one can generate the S-shaped curvewith theM/G/c/c model, then identify the point-of-inflection fromthe curve itself.


Fig. 11. Kerner’s empirical travel times around an incident in a road.

Also, from a practical perspective, knowing that the point-of-inflection of the travel time function corresponds to theaverage travel time T and to the maximum throughput θmax means that one can readily compute the travel time valuesalong a path using Little’s Law [36] as:

T =Ls

θmax, (19)

Tmax =2Lsθmax

, (20)

where Ls is the average number of vehicles along the path. Eqs. (19) and (20) are very compact formulas that depend uponknowing Ls, but that is possible from the M/G/c/c software tools or from fitting the logistic equation (18) to sample trafficflow data.

Also, one might argue that the travel time function is not only for one segment of road traffic but a network ofM/G/c/c queues. However, the travel time function can be generated from anM/G/c/c networkmodel of traffic flows. Thismethodology for computing theM/G/c/c network performance measures is discussed in several related papers [43,45].

4.4. Empirical evidence of S-shaped travel time curve

As empirical evidence and verification of the S-shaped travel time model, Kerner [1] did a study of vehicle travel timesaround an incident where a floating car vehicle was used to record travel times in and around the incident. The graph ofthe empirical travel times is indicated in Fig. 11 which is modeled after the graph appearing in Kerner et al. [46]. Clearly,this represents an S-shaped curve where the three-phases are as earmarked. Fig. 11 also includes the dissolution of thecongestion and the return to normal free-flow travel time.

5. Examples

If we examine some incidents and bottlenecks (on-ramps) with the M/G/c/c model, we can also compare theM/G/c/c approach with the other travel time models when these problems occur. As we shall argue, this will further becorroborated with Kerner’s empirical work.

More importantly, in utilizing the simulationmodel as opposed to the steady-stateM/G/c/c model, we can demonstratethe nonlinear oscillatory behavior of the bottleneck or incident on the traffic flow travel time dynamics.

5.1. Incidence model

We are interested in utilizing M/G/c/c models for modeling incidences vehicular traffic systems. A basic model forcapturing the phenomenon of traffic incidence is given in Fig. 12, where for a given traffic segment, the flow of vehiclesis impeded by the reduction in capacity of the highway segment. The top half of Fig. 12 represents a traffic segment whichis unimpeded while the bottom half represents the reduction in capacity due to some incident. The incident will cause amerging of the two flows (part 1) at the point of incidence (part 2) where then blocking of the traffic flows will occur as afunction of the volume of traffic arrival processes. Once the traffic emerges from the end of the incident (part 3), the trafficwill proceed to even out into the two lanes. Five nodes are necessary to capture the traffic interruption due to the incidence,so the problem becomes quite complex. The situation sketched can be understood as an increase in local variability as the


Fig. 12. Basic incidence model for a given traffic segment where the flow is impeded by the reduction in capacity of the highway segment.

Table 2Analytical and simulated travel times for basic incidence model.

λ Transition area Incident area Termination area ttotalt1 t2 t3 SimulationBPR M/G/c/c BPR M/G/c/c BPR M/G/c/c BPR M/G/c/c Mean SE mean cpu

500 0.016 0.017 0.016 0.019 0.016 0.017 0.048 0.054 0.054 0.0001 0 h 0 m 2s1000 0.016 0.019 0.016 0.021 0.016 0.019 0.048 0.059 0.059 0.0001 0 h 0 m 7s1500 0.016 0.020 0.016 0.025 0.016 0.020 0.048 0.065 0.064 0.0002 0 h 0 m 15s2000 0.016 0.021 0.017 0.029 0.016 0.021 0.049 0.072 0.072 0.0003 0 h 0 m 27s2500 0.016 0.076 0.021 0.038 0.016 0.023 0.053 0.137 0.084 0.001 0 h 0 m 49s3000 0.016 0.129 0.046 0.064 0.016 0.024 0.078 0.217 0.228 0.001 1 h 25 m 48s3500 0.016 0.138 0.155 0.069 0.016 0.024 0.187 0.231 0.233 0.001 1 h 36 m 35s

two flows need to merge and therefore will take more space per vehicle just before the incident; just after the incident thereverse phenomenon takes place.

We provide a sample of results of the incident. First, with the analytical models of the process (BPR andM/G/c/c), thenanotherwith a simulationmodel of this process. Let us assume that the traffic process is first a highwaywhere traffic arrivalsoccur at λ for both lanes. The traffic segments are divided into three 1-mile-long segments, and when the middle segmentloses capacity, the merge node is 1-lane wide. Otherwise, the nodes are 2-lane wide.

In Table 2, we present results for the basic network model for analytical and simulation approaches. It is noticeable thatunder low arrival rates both analytical methods (BPR andM/G/c/c) seems to agree but the differences become large whenthe traffic is heavy. The M/G/c/c model is also compared with simulations and the results are close and mostly within thestandard error of means. Regarding the simulation model, we remark that the processing times are really very high as thetraffic becomes heavy and, consequently, the number of vehicle entities in the simulation model explodes along with theCPU times.

Fig. 13 illustrates the S-shaped curve of the incidencemodel for theM/G/c/c travel time function alongwith the one fromthe BPR function and the simulation model. Notice that the BPR function greatly underestimates the congestion problemsat the incident until the traffic is extremely heavy.

As a footnote to the above type of study, recently, many states in the USA have begun to require queueing analysis studiesof highway work zones in order to estimate the travel delay and queueing length effects of these traffic flow interruptions.Not only for safety reasons, but for travel planning, construction, and overall management of the traffic incident, queueingstudies are crucial.

5.2. Oscillatory model

Finally, in a related measure of the M/G/c/c model for modeling vehicular traffic flows, a small study of an on-rampis proposed. What is intended to show here is that through the transient M/G/c/c model, that the oscillatory behavior of


Fig. 13. Travel times for the BPR function, theM/G/c/c model, and the simulation model as functions of the arrival rate (λ) for the basic incidence model(Fig. 12).

Fig. 14. A three-node merge with one lane wide and one mile long representing an on ramp situation.

traffic in the vicinity of an incident can be shown. This also corresponds to the oscillatory behavior of empirically based trafficflows that Kerner et al. [46] and Jiang et al. [47] have shown. Davis [48] has also demonstrated the oscillatory behavior ata highway bottleneck using a different type of simulation model than the M/G/c/c one described in this paper which alsorelies on the three-phase microscopic model of Kerner and Klenov [49–52] made in the framework of Kerner’s theory.

For the example, in Fig. 14, an on ramp situation, actually, a three-node merge, with one lane wide and one mile longeach is considered. The situation here is similar to the case presented in Fig. 12, i.e., a locally increase in the variability astwo flows need to merge.

In Fig. 15, it is observed that when the arrival rate is moderate (500 and 1000 vehicles/h) the number of vehicles exitedis approximately stable and centered around the arrival rate. However, for heavy traffic, the number of exiting vehiclesoscillates. Additionally, above 2000 vehicles/h, the number of vehicles exited does not increase any longer, indicating thesystem saturation (wide-moving jam). These theoretical results correspond to the empirical results of Kerner [1] (Figure 9.6,p. 252), and the results of Davis [48] in and around a bottleneck.

This type of oscillatory behavior for the exiting traffic occurs in other particle or state dependent models withM/G/c/c queues and queueing networks, such as pedestrian flows in and around a bottleneck.

5.3. Open questions and future research

There are a number of directions possible with this research. One is to recognize that theM/G/c/c model is also directlyapplicable to modeling pedestrian networks [28], so many of the similar features of the travel delay function as shown inthis paper apply to pedestrian dynamics. Some of the research already conducted for pedestrian networks has occurred inMitchell and MacGregor Smith [45] and MacGregor Smith [53–55].

One can also use this model in traffic assignment applications and the authors are considering the design of an algorithmfor such an application in future papers [5]. The difficulty with this traffic assignment algorithm, however, is that the traveltime function is non-convex, so care must be taken in designing the nonlinear programming methodology.


Fig. 15. The oscillatory behavior of the number of vehicles exited (throughput) for several arrival rates (λ = 500, 1000, 2000, and 4000 vehicles/h) for theon-ramp configuration (Fig. 14).

6. Summary and conclusions

This paper has presented an overview of the problem of estimating travel time on road links. Three different modelswere examined for their ability to estimate the travel time over road links under various situations. It has been shown thatthe theoretical M/G/c/c models support the empirically-based three-phase traffic flow model of Kerner [1]. The S-shapedcurves of theM/G/c/c state-dependent model are felt to be an important contribution to the understanding of travel-timebehavior, since they provide a quantitative way to define the maximum flow rate of a highway segment.

Acknowledgments

The second author was partially funded by Brazilian Agencies CNPq, FAPEMIG, and PRPq-UFMG.

Appendix. Tables

See Tables 3 and 4.

Table 3Analytical and simulated performance measures versus arrival rate (linear).

L λ pc θ E(N) E(T ) *cpu (s)Anlt. Simulation∗ Anlt. Simulation∗ Anlt. Simulation∗ Anlt. Simulation∗

Average 95% CI Average 95% CI Average 95% CI Average 95% CI

1 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 8.35 8.31 [8.27; 8.35] 0.017 0.017 [0.017; 0.017] 3.91000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 17.5 17.6 [17.5; 17.6] 0.018 0.018 [0.018; 0.018] 121500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 27.9 27.8 [27.7; 27.9] 0.019 0.019 [0.019; 0.019] 242000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 40.1 40.0 [39.9; 40.2] 0.020 0.020 [0.020; 0.020] 422500 0.974 0.000 [0.000; 0.000] 64.2 2497 [2492; 2503] 200 55.6 [55.4; 55.8] 3.12 0.022 [0.022; 0.022] 703000 0.979 0.979 [0.978; 0.979] 63.9 64.1 [63.5; 64.8] 200 201 [199; 203] 3.13 3.13 [3.13; 3.13] 503500 0.982 0.982 [0.982; 0.982] 63.7 62.5 [62.3; 62.7] 200 196 [196; 197] 3.14 3.14 [3.14; 3.14] 51

2 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 16.7 16.6 [16.5; 16.7] 0.033 0.033 [0.033; 0.033] 5.61000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 35.1 35.1 [35.0; 35.3] 0.035 0.035 [0.035; 0.035] 191500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 55.8 55.6 [55.4; 55.8] 0.037 0.037 [0.037; 0.037] 41

(continued on next page)


Table 3 (continued)



2000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 80.1 79.9 [79.6; 80.2] 0.040 0.040 [0.040; 0.040] 772500 0.987 0.000 [0.000; 0.000] 31.7 2497 [2492; 2503] 400 111 [110; 111] 12.6 0.044 [0.044; 0.044] 1423000 0.989 0.884 [0.786; 0.982] 31.6 347 [53.2; 640] 400 279 [239; 319] 12.7 7.69 [6.03; 9.35] 1583500 0.991 0.991 [0.991; 0.991] 31.5 32.3 [31.9; 32.6] 400 392 [388; 397] 12.7 12.2 [12.1; 12.2] 112

5 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 41.7 41.6 [41.3; 41.8] 0.083 0.083 [0.083; 0.083] 101000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 87.7 87.9 [87.5; 88.2] 0.088 0.088 [0.088; 0.088] 421500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1501] 139 139 [139; 140] 0.093 0.093 [0.093; 0.093] 1072000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 200 200 [199; 200] 0.100 0.100 [0.100; 0.100] 2272500 0.995 0.000 [0.000; 0.000] 12.6 2497 [2492; 2503] 1000 276 [275; 277] 79.6 0.111 [0.111; 0.111] 4233000 0.996 0.000 [0.000; 0.000] 12.6 3000 [2995; 3005] 1000 405 [402; 407] 79.7 0.135 [0.134; 0.135] 7523500 0.996 0.998 [0.997; 0.998] 12.5 8.45 [8.11; 8.78] 1000 124 [119; 129] 79.7 14.7 [14.6; 14.8] 281

10 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 83.5 83.1 [82.7; 83.5] 0.167 0.167 [0.167; 0.167] 201000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 175 176 [175; 176] 0.175 0.175 [0.175; 0.175] 961500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 279 278 [277; 279] 0.186 0.186 [0.186; 0.186] 2572000 0.000 0.000 [0.000; 0.000] 2000 1995 [1990; 2001] 400 399 [398; 400] 0.200 0.200 [0.200; 0.200] 5012500 0.997 0.000 [0.000; 0.000] 6.27 2498 [2493; 2503] 2000 552 [551; 554] 319 0.221 [0.221; 0.221] 9003000 0.998 0.000 [0.000; 0.000] 6.26 3000 [2995; 3005] 2000 803 [799; 807] 319 0.268 [0.267; 0.269] 15193500 0.998 0.999 [0.999; 0.999] 6.26 3.78 [3.60; 3.97] 2000 53.5 [51.0; 56.1] 319 14.2 [14.0; 14.3] 605

Table 4Analytical and simulated performance measures versus arrival rate (exponential).



1 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 9.35 9.30 [9.24; 9.35] 0.019 0.019 [0.019; 0.019] 6.91000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 21.3 21.4 [21.3; 21.5] 0.021 0.021 [0.021; 0.021] 191500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 36.9 36.7 [36.6; 36.9] 0.025 0.025 [0.025; 0.025] 402000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 58.6 58.4 [58.1; 58.7] 0.029 0.029 [0.029; 0.029] 732500 0.000 0.000 [0.000; 0.000] 2500 2497 [2492; 2503] 95.0 94.7 [94.2; 95.3] 0.038 0.038 0.038; 0.038] 1343000 0.052 0.051 [0.050; 0.053] 2843 2843 [2842; 2844] 183 183 [182; 183] 0.064 0.064 [0.064; 0.064] 3133500 0.188 0.187 [0.186; 0.189] 2841 2841 [2841; 2841] 196 196 [196; 196] 0.069 0.069 [0.069; 0.069] 354

2 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 18.6 18.5 [18.4; 18.6] 0.037 0.037 [0.037; 0.037] 8.91000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1005] 42.4 42.4 [42.2; 42.6] 0.042 0.042 [0.042; 0.042] 281500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 73.2 72.9 [72.6; 73.2] 0.049 0.049 [0.049; 0.049] 642000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 116 115 [115; 116] 0.058 0.058 [0.058; 0.058] 1302500 0.000 0.000 [0.000; 0.000] 2500 2497 [2492; 2503] 186 186 [185; 187] 0.075 0.074 [0.074; 0.075] 2803000 0.055 0.054 [0.052; 0.055] 2836 2836 [2836; 2837] 382 382 [381; 383] 0.135 0.135 [0.134; 0.135] 7133500 0.191 0.190 [0.189; 0.192] 2830 2830 [2830; 2830] 396 396 [396; 396] 0.140 0.140 [0.140; 0.140] 754

5 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 46.5 46.2 [46.0; 46.5] 0.093 0.093 [0.093; 0.093] 151000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1006] 106 106 [105; 106] 0.106 0.106 [0.105; 0.106] 591500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1501] 182 181 [181; 182] 0.121 0.121 [0.121; 0.121] 1652000 0.000 0.000 [0.000; 0.000] 2000 1996 [1990; 2001] 288 287 [285; 288] 0.144 0.144 [0.143; 0.144] 3742500 0.000 0.000 [0.000; 0.000] 2500 2498 [2492; 2503] 461 460 [457; 463] 0.184 0.184 [0.184; 0.185] 7573000 0.058 0.057 [0.055; 0.059] 2826 2827 [2826; 2827] 983 983 [982; 985] 0.348 0.348 [0.348; 0.348] 17553500 0.193 0.192 [0.191; 0.194] 2823 2823 [2823; 2824] 996 996 [996; 996] 0.353 0.353 [0.353; 0.353] 1852

10 500 0.000 0.000 [0.000; 0.000] 500 498 [495; 500] 92.8 92.4 [91.9; 92.9] 0.186 0.186 [0.185; 0.186] 261000 0.000 0.000 [0.000; 0.000] 1000 1002 [998; 1006] 211 211 [210; 212] 0.211 0.211 [0.211; 0.211] 1361500 0.000 0.000 [0.000; 0.000] 1500 1496 [1492; 1500] 363 362 [361; 363] 0.242 0.242 [0.242; 0.242] 3602000 0.000 0.000 [0.000; 0.000] 2000 1996 [1991; 2001] 574 572 [569; 574] 0.287 0.287 [0.286; 0.287] 7452500 0.000 0.000 [0.000; 0.000] 2500 2498 [2492; 2503] 919 916 [911; 922] 0.368 0.367 [0.366; 0.368] 14583000 0.059 0.058 [0.057; 0.060] 2822 2822 [2821; 2823] 1984 1984 [1983; 1985] 0.703 0.703 [0.703; 0.704] 36913500 0.194 0.193 [0.191; 0.194] 2820 2821 [2821; 2822] 1996 1996 [1996; 1996] 0.708 0.708 [0.707; 0.708] 3861

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